A daredevil on a motorcycle leaves the end of a ramp with a speed of 29.5 m/s as in the figure below. If his speed is 27.1 m/s when he reaches the peak of the path, what is the maximum height that he reaches? Ignore friction and air resistance.
Well I do not see your picture but if energy is conserved
m g (increase in height) = (m/2)(29.5^2-27.1^2)
mass cancels
increase in height =(29.5^2-27.1^2)/(2g)
To solve this problem, we can use the principle of conservation of mechanical energy. At the top of the path, the daredevil's kinetic energy will be completely converted into gravitational potential energy.
The equation for mechanical energy is as follows:
Mechanical Energy = Kinetic Energy + Potential Energy
Where:
Mechanical Energy = Total mechanical energy of the system
Kinetic Energy = 0.5 * mass * velocity^2
Potential Energy = mass * gravitational acceleration * height
In this case, we ignore the mass of the daredevil as it cancels out in the equation.
At the start of the ramp, the daredevil's kinetic energy is:
Kinetic Energy = 0.5 * (velocity)^2
= 0.5 * (29.5)^2
= 431.375 J
At the top of the motion, the daredevil's velocity is 27.1 m/s. So his kinetic energy at the peak is:
Kinetic Energy = 0.5 * (velocity)^2
= 0.5 * (27.1)^2
= 368.71 J
Since there is no friction or air resistance, the mechanical energy is conserved. Thus, we can equate the initial kinetic energy to the final potential energy:
431.375 J = gravitational potential energy at the peak
Simplifying, we find:
Gravitational Potential Energy = 431.375 J
Now, we can solve for the height (h) using the equation for gravitational potential energy:
Potential Energy = mass * gravitational acceleration * height
Since we are ignoring the mass, we can simplify the equation to:
Potential Energy = gravitational acceleration * height
Rearranging the equation, we find:
height = Potential Energy / gravitational acceleration
Substituting the given values, we have:
height = 431.375 J / 9.8 m/s^2
Calculating the height, we get:
height = 44.02 meters
Therefore, the daredevil reaches a maximum height of 44.02 meters.
To determine the maximum height reached by the daredevil, we can use the principle of conservation of mechanical energy. In the absence of friction and air resistance, mechanical energy is conserved throughout the motion.
The mechanical energy of the daredevil at the bottom of the ramp consists of two components: kinetic energy (KE) and gravitational potential energy (PE).
At the bottom of the ramp:
KE1 + PE1 = Total Mechanical Energy (E1)
At the top of the path (maximum height):
KE2 + PE2 = Total Mechanical Energy (E2)
Given that the daredevil's speed at the bottom of the ramp (v1) is 29.5 m/s and his speed at the top of the path (v2) is 27.1 m/s, we can determine the maximum height he reaches by equating the mechanical energies at the bottom and top.
So, the equation becomes:
(1/2)mv1^2 + mgh1 = (1/2)mv2^2 + mgh2
where m is the mass of the daredevil, g is the acceleration due to gravity, h1 is the height at the bottom of the ramp (which is zero), and h2 is the maximum height we want to find.
Since h1 is zero, the equation simplifies to:
(1/2)mv1^2 = (1/2)mv2^2 + mgh2
Simplifying further by canceling out m and dividing everything by (1/2), we get:
v1^2 = v2^2 + 2gh2
Substituting the given values, the equation becomes:
29.5^2 = 27.1^2 + 2gh2
Now, we can solve for h2:
h2 = (29.5^2 - 27.1^2) / (2g)
To find the value of g, which is the acceleration due to gravity, we can use the standard value of 9.8 m/s^2.
Substituting the values, we have:
h2 = (29.5^2 - 27.1^2) / (2 * 9.8)
Calculating this expression will give us the maximum height reached by the daredevil.